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This is my attempt to create a brute force algorithm that can use any hash or encryption standard. Bruteforcing has been around for some time now, but it is mostly found in a pre-built application that performs only one function. My attempt to bruteforcing started when I forgot a password to an archived rar file.
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'Proof by cases' redirects here. For the concept in propositional logic, see.Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of. A proof by exhaustion typically contains two stages:. A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. A proof of each of the cases.The prevalence of digital has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of in 1976), though such approaches can also be challenged on the basis of.
Can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.In the, proof by exhaustion and case analysis are related to ML-style. Contents.Example Proof by exhaustion can be used to prove that if an is a, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9.Proof:Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three cases are exhaustive:. Case 1: If n = 3 p, then n 3 = 27 p 3, which is a multiple of 9. Case 2: If n = 3 p + 1, then n 3 = 27 p 3 + 27 p 2 + 9 p + 1, which is 1 more than a multiple of 9.
For instance, if n = 4 then n 3 = 64 = 9×7 + 1. Case 3: If n = 3 p − 1, then n 3 = 27 p 3 − 27 p 2 + 9 p − 1, which is 1 less than a multiple of 9.
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For instance, if n = 5 then n 3 = 125 = 9×14 − 1.Elegance Mathematicians prefer to avoid proofs by exhaustion with large numbers of cases, which are viewed as. An illustration as to how such proofs might be inelegant is to look at the following proofs that all modern are held in years which are divisible by 4:Proof: The were held in 1896, and then every 4 years thereafter (neglecting years in which the games were not held due to World War I and World War II). Since 1896 = 474 × 4 is divisible by 4, the next Olympics would be in year 474 × 4 + 4 = (474 + 1) × 4, which is also divisible by four, and so on (this is a proof by ). Therefore the statement is proved.The statement can also be proved by exhaustion by listing out every year in which the Summer Olympics were held, and checking that every one of them can be divided by four. With 28 total Summer Olympics as of 2016, this is a proof by exhaustion with 28 cases.In addition to being less elegant, the proof by exhaustion will also require an extra case each time a new Summer Olympics is held. This is to be contrasted with the proof by mathematical induction, which proves the statement indefinitely into the future.Number of cases There is no upper limit to the number of cases allowed in a proof by exhaustion.
Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving a might involve considering a very large number of possible positions in the of that problem.The first proof of the was a proof by exhaustion with 1,936 cases.
This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.In general the probability of an error in the whole proof increases with the number of cases. A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection.
Other types of proofs—such as proof by induction —are considered more. However, there are some important theorems for which no other method of proof has been found, such as. The proof that there is no finite of order 10. The. The. The.See also.Notes.
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